Question

If $N$ is the number of positive integral solutions of $x_1\times x_2\times x_3\times x_4=770$, then:

• A. $N$ is divisible by $4$ district primes
• B. $N$ is a perfect square
• C. $N$ is a perfect fourth power
• D. $N$ is a perfect $8^{th}$ power

Answer 1

Answer: (B), (C), (D)

The correct options are
B $N$ is a perfect square
C $N$ is a perfect fourth power
D $N$ is a perfect $8^{th}$ power

(B, C, D) $770=\left(2\right)\left(5\right)\left(7\right)\left(11\right).$ We can assign each of $2,5,7,1$ in $4$ ways. Thus, required number of solutions
$4^4=256={16}^2=2^8$